For some reason it, I need say "I don't have any toothpaste", instead of simply saying "I don't have toothpaste"
Because 'toothpaste' is a mass noun, maybe? ;-)
Is zębów here a plural? And what grammatical case is it in?
Just so you know, „do” is actually a fairly dependant preposition in that it always takes genitive – no differentiation for place or time or anything; if you see „do” you always know genitive is what follows. ;)
Good cheat sheet about those on Wikibooks
I do not think the "pasty", in this case, is plural. I believe it is "pasty" because the negated accusative verb (mice) takes on genitive and genitive for "pasta" is "pasty". Is this correct?
Yes, you're right. But that comment was about "zębów" :)
I'm confused by 'mice', I guess it's a typo plus lack of Polish signs of "mieć"?
Why does Duo reject my correct "I haven't any toothpaste.", instead insisting on the colloquial "I haven't got toothpaste."?
"Got" is OK in spoken UK English, but shouldn't really be written.
In US English, "gotten" would probably be preferred...
Added "I haven't any toothpaste".
I'd say that we in Poland are generally taught that "we have got" is a perfectly fine form, although not as basic as simple "we have".
I believe that 'gotten' would change the meaning...